topology a geometric approach jun 2003

Topology: A Geometric Approach Terry Lawson Mathematics Department, Tulane University, New Orleans, LA 70118 1 Oxford Graduate Texts in Mathematics Series Editors R. Cohen S. K. Donaldson S. Hildebrandt T. J. Lyons M. J. Taylor OXFORD GRADUATE TEXTS IN MATHEMATICS 1. Keith Hannabuss: An Introduction to Quantum Theory 2. Reinhold Meise and Dietmar Vogt: Introduction to Functional Analysis 3. James G. Oxley: Matroid Theory 4. N.J. Hitchin, G.B. Segal, and R.S. Ward: Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces 5. Wulf Rossmann: Lie Groups: An Introduction Through Linear Groups 6. Q. Liu: Algebraic Geometry and Arithmetic Curves 7. Martin R. Bridson and Simon M, Salamon (eds): Invitations to Geometry and Topology 8. Shmuel Kantorovitz: Introduction to Modern Analysis 9. Terry Lawson: Topology: A Geometric Approach 10. Meinolf Geck: An Introduction to Algebraic Geometry and Algebraic Groups 3 Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c  Oxford University Press, 2003 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2003 First published in paperback 2006 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Lawson, Terry, 1945– Topology : a geometric approach / Terry Lawson. (Oxford graduate texts in mathematics ; 9) Includes bibliographical references and index. 1. Topology. I. Title. II. Series. QA611.L36 2003 514–dc21 2002193104 Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd, King’s Lynn ISBN 0–19–851597–9 978–0–19–851597–5 ISBN 0–19–920248–6 (Pbk.) 978–0–19–920248–5 (Pbk.) 13579108642 Preface This book is intended to introduce advanced undergraduates and beginnning graduate students to topology, with an emphasis on its geometric aspects. There are a variety of influences on its content and structure. The book consists of two parts. Part I, which consists of the first three chapters, attempts to provide a balanced view of topology with a geometric emphasis to the student who will study topology for only one semester. In particular, this material can provide undergraduates who are not continuing with graduate work a capstone exper- ience for their mathematics major. Included in this experience is a research experience through projects and exercise sets motivated by the prominence of the Research Experience for Undergraduate (REU) programs that have become important parts of the undergraduate experience for the best students in the US as well as VIGRE programs. The book builds upon previous work in real analysis where a rigorous treatment of calculus has been given as well as ideas in geometry and algebra. Prior exposure to linear algebra is used as a motiv- ation for affine linear maps and related geometric constructions in introducing homeomorphisms. In Chapter 3, which introduces the fundamental group, some group theory is developed as needed. This is intended to be sufficient for students without a prior group theory course for most of Chapter 3. A prior advanced undergraduate level exposure to group theory is useful for the discussion of the Seifert–van Kampen theorem at the end of Chapter 3 and for Part II. Part I provides enough material for a one-semester or two-quarter course. In these chapters, material is presented in three related ways. The core of these chapters presents basic material from point set topology, the classification of surfaces, and the fundamental group and its applications with many details left as exercises for the student to verify. These exercises include steps in proofs as well as application of the theory to related examples. This style fosters the highly involved approach to learning through discussion and student presenta- tion which the author favors, but also allows instructors who prefer a lecture approach to include some of these details in their presentation and to assign others. The second method of presentation comes from chapter-end exercise sets. Here the core material of the chapter is extended significantly. These exercise sets include material an instructor may choose to integrate as additional topics for the whole class, or they may be used selectively for different types of students to individualize the course. The author has used them to give graduate students and undergraduates in the same course different types of assignments to assure vi Preface that undergraduates get a well-balanced exposure to topology within a semester while graduate students get exposure to the required material for their PhD written examinations. Finally, these chapters end with a project, which provides a research experience that draws upon the ideas presented in the chapter. The author has used these projects as group projects which lead to the students involved writing a paper and giving a class presentation on their project. Part II, which consists of Chapters 4–6, extends the material in a way to make the book useful as well for a full-year course for first-year graduate students with no prior exposure to topology. These chapters are written in a very different style, which is motivated in part by the ideal of the Moore method of teaching topology combined with ideas of VIGRE programs in the US which advocate earlier introduction of seminar and research activities in the advanced under- graduate and graduate curricula. In some sense, they are a cross between the chapter-end exercises and the projects that occur in Part I. These last chapters cover material from covering spaces, CW complexes, and algebraic topology through carefully selected exercise sets. What is very different from a pure Moore method approach is that these exercises come with copious hints and suggested approaches which are designed to help students master this material while at the same time improving their abilities in understanding the structure of the subject as well as in constructing proofs. Instructors may use them with a teach- ing style which ranges from a pure lecture–problem format, where they supply key proofs, to a seminar–discussion format, where the students do most of the work in groups or individually. Class presentations and expository papers by students, in groups or individually, can also be a component here. The goal is to lay out the basic structure of the material in carefully developed problem sets in a way that maximizes the flexiblity of the instructor in utilizing this material and encourages strong involvement of students in learning it. We briefly outline what is covered in the text. Chapter 1 gives a basic intro- duction to the point set topology used in the rest of the book, with emphasis on developing a geometric feel for the concepts. Quotient space constructions of spaces built from simpler pieces such as disks and rectangles is stressed as it is applied frequently in studying surfaces. Chapter 2 gives the classification of sur- faces using the viewpoint of handle decompositions. It provides an application of the ideas in the first chapter to surface classification, which is an important example for the whole field of manifold theory and geometric topology. Chapter 3 introduces the fundamental group and applies it to many geometric problems, including the final step in the classification of surfaces of using it to distin- guish nonhomeomorphic surfaces. In Chapter 3, certain basic ideas of covering spaces (particularly that of exponential covering of the reals over the circle) are used, and Chapter 4 is concerned with developing these further into the beauti- ful relationship between covering spaces and the fundamental group. Chapter 5 discusses CW complexes, including simplicial complexes and ∆-complexes. CW complexes are motivated by earlier work from handle decompositions and used later in studying homology. Chapter 6 gives a selective approach to homology the- ory with emphasis on its application to low-dimensional examples. In particular, Preface vii it gives the proof (through exercise sets) of key results such as invariance of domain and the Jordan curve theorem which were used earlier. It also gives a more advanced approach to the concept of orientation, which plays a key role in Chapter 2. The coverage in the text differs substantially in content, order, and type from texts at a similar level. The emphasis on geometry and the desire to have a balanced one-semester introduction leads to less point set topology but a more thorough application of it through the handlebody approach to surface classific- ation. We also move quickly enough to allow a significant exposure to algebraic topology through the fundamental group within the first semester. The extens- ive exercise sets, which form the core of developing the more advanced material in the text, also foster more flexibility in how the text can be used. When indi- vidual parts are counted, there are more than a thousand exercises in the text. In particular, it should serve well as a resource for independent study and projects outside of the standard course structure as well as allow many different types of courses. There is an emphasis on understanding the topology of low-dimensional spaces which exist in three-space, as well as more complicated spaces formed from planar pieces. This particularly occurs in understanding basic homotopy theory and the fundamental group. Because of this emphasis, illustrations play a key role in the text. These have been prepared with LaTeX tools pstricks and xypic as well as using figures constructed using Mathematica, Matlab, and Adobe Illustrator. The material here is intentionally selective, with the dual goals of first giv- ing a good one-semester introduction within the first three chapters and then extending this to provide a problem-oriented approach to the remainder of a year course. We wish to comment on additional sources which we recommend for material not covered here, or different approaches to our material where there is overlap. For a more thorough treatment of point set topology, we recommend Munkres [24]. For algebraic topology, we recommend Hatcher [13] and Bredon [5]. All of these books are written at a more advanced level than this one. We have used these books in teaching topology at the first- and second-year graduate levels and they influenced our approach to many topics. For some schools with strong graduate students, it may be most appropriate to use just the first three chapters of our text for undergraduates and to prepare less prepared graduate students for the graduate course on the level of one of the three books mentioned. In that situation, some of the projects or selected exercises from Chapters 4–6 could be used as enhancements for the graduate course. The book contains as an appendix some selected solutions to exercises to assist students in learning the material. These solutions are limited in number in order to maximize the flexibility of instructors in using the exercise sets. Instruct- ors who are adopting this book for use in a course can obtain an Instructor Solutions Manual with solutions to the exercises in the book in terms of a PDF file through Oxford University Press (OUP). The LaTeX files for these solutions are also available through OUP for those instructors who wish to use them in viii Preface preparation of materials for their class. Please write to the following address, and include your postal and e-mail addresses and full course details including student numbers: Marketing Manager Mathematics and Statistics Academic and Professional Publishing Oxford University Press Great Clarendon Street Oxford OX2 6DP, UK Contents List of Figures xi I A Geometric Introduction to Topology 1 Basic point set topology 3 1.1 Topology in R n 3 1.2 Open sets and topological spaces 7 1.3 Geometric constructions of planar homeomorphisms 15 1.4 Compactness 22 1.5 The product topology and compactness in R n 26 1.6 Connectedness 30 1.7 Quotient spaces 37 1.8 The Jordan curve theorem and the Sch¨onflies theorem 44 1.9 Supplementary exercises 49 2 The classification of surfaces 62 2.1 Definitions and construction of the models 62 2.2 Handle decompositions and more basic surfaces 68 2.3 Isotopy and attaching handles 77 2.4 Orientation 88 2.5 Connected sums 98 2.6 The classification theorem 106 2.7 Euler characteristic and the identification of surfaces 119 2.8 Simplifying handle decompositions 126 2.9 Supplementary exercises 133 3 The fundamental group and its applications 153 3.1 The main idea of algebraic topology 153 3.2 The fundamental group 160 3.3 The fundamental group of the circle 167 3.4 Applications to surfaces 172 3.5 Applications of the fundamental group 179 3.6 Vector fields in the plane 185 x Contents 3.7 Vector fields on surfaces 194 3.8 Homotopy equivalences and π 1 206 3.9 Seifert–van Kampen theorem and its application to surfaces 215 3.10 Dependence on the base point 226 3.11 Supplementary exercises 230 II Covering Spaces, CW Complexes and Homology 4 Covering spaces 243 4.1 Basic examples and properties 243 4.2 Conjugate subgroups of π 1 and equivalent covering spaces 248 4.3 Covering transformations 254 4.4 The universal covering space and quotient covering spaces 256 5 CW complexes 260 5.1 Examples of CW complexes 260 5.2 The Fundamental group of a CW complex 266 5.3 Homotopy type and CW complexes 269 5.4 The Seifert–van Kampen theorem for CW complexes 275 5.5 Simplicial complexes and ∆-complexes 276 6 Homology 281 6.1 Chain complexes and homology 281 6.2 Homology of a ∆-complex 283 6.3 Singular homology H i (X) and the isomorphism π ab 1 (X, x) ≃ H 1 (X) 286 6.4 Cellular homology of a two-dimensional CW complex 292 6.5 Chain maps and homology 294 6.6 Axioms for singular homology 300 6.7 Reformulation of excision and the Mayer–Vietoris exact sequence 304 6.8 Applications of singular homology 308 6.9 The degree of a map f : S n → S n 310 6.10 Cellular homology of a CW complex 313 6.11 Cellular homology, singular homology, and Euler characteristic 320 6.12 Applications of the Mayer–Vietoris sequence 323 6.13 Reduced homology 328 6.14 The Jordan curve theorem and its generalizations 329 6.15 Orientation and homology 333 6.16 Proof of homotopy invariance of homology 345 6.17 Proof of the excision property 350 Appendix Selected solutions 355 References 383 Index 385 [...]... v 1 = a1 − a0 , v 2 = a2 − a0 The composition of multiplication by V and the translation then gives a map, called an a ne linear map, which is a homeomorphism between the standard triangle ∆(e0 , e1 , e2 ) and ∆ (a0 , a1 , a2 ) This a ne linear map A has the property that λ0 e0 + λ1 e1 + λ2 e2 is sent to λ0 a0 + λ1 a1 + λ2 a2 In particular, this means that the triangle ∆ (a0 , a1 , a2 ) is characterized... canonical a ne linear map, and the image of a triangle under an a ne linear map will be another triangle In particular, there is no a ne linear map sending a triangle to a rectangle A ne linear maps from one triangle ∆ (a0 , a1 , a2 ) to another triangle ∆(b0 , b1 , b2 ) are determined completely by the map on the vertices ai → bi and the a ne linearity condition λi ai → λi ai Exercise 1.3.3 (a) Show that... is a translation and L is a linear map (d) Show that any composition M of translations and linear maps satisfies k k k M ( i=1 λi ai ) = i=1 λi M (ai ) when i=1 λi = 1 Conversely, show that if M satisfies this condition for any three a nely independent points, then M is a composition of a translation and a linear map, so is an a ne linear map (e) Show that an a ne linear map sending ai to bi will always... v 1 = a1 − a0 , v 2 = a2 − a0 are linearly independent, then we say that a0 , a1 , a2 are a nely independent This is equivalent to λ1 a0 + λ1 a1 + λ2 a2 = 0, λ0 + λ1 + λ2 = 0 implying λ0 = λ1 = λ2 = 0 If a0 , a1 , a2 are a nely independent, then there is a triangle ∆ (a0 , a1 , a2 ) with vertices a0 , a1 , a2 Translation by a0 gives a homeomorphism between ∆(e0 , v 1 , v 2 ) and ∆ (a0 , a1 , a2 ), where... int A, is the union of all open sets contained in A A point in int A is called an interior point of A The ¯ boundary of A, denoted Bd A, is A ∩ X \A A point in Bd A is called a boundary point of A ¯ Exercise 1.2.11 Show that A is closed and int A is open 1.2 Open sets and topological spaces 13 ¯ To find A in examples, it is useful to have another characterization Note ¯ that a point x is not in A exactly... 1.3.1 and the inequality |Ax − Ay| ≤ A |x − y| shown in linear algebra Note that a rotation is reversible; after rotating a point by an angle θ, we can get back to our original point by rotating by an angle −θ From the matrix point of view, the matrix A is invertible Either way may be used to show that rotation represents a homeomorphism from the plane to itself Another familiar geometric operation... triangles A1 , A2 , the map is a ne linear ( i λi ai → i λi bi ) Our homeomorphism is an example of a piecewise linear (PL) homeomorphism of planar regions—the domain and range are divided into triangles, and the homeomorphism is an a ne linear homeomorphism on each triangle We can generalize the argument above to show that any two convex polygonal regions in the plane are homeomorphic By a polygonal... continuous That a rotation does in fact preserve distances can be checked using trigonometric formulas and the distance formula in the plane Another way of seeing that a rotation by φ is continuous is to note that it is given by a linear map, x → Ax, where x represents a point in the plane as a column vector and A is the 2 × 2 matrix cos φ − sin φ sin φ cos φ For a rotation, A is an orthogonal matrix, which... was necessary as part of the rigid motion We have seen that congruences and similarities are both examples of homeomorphisms In geometry, a triangle and a rectangle are distinguished from one another by the number of sides, and two triangles, although possibly not similar, still are seen to have the same “shape” We will see below that the inside of a triangle and the inside of a rectangle are in fact... 2.7 2.8 Balls are open Open and closed rectangles Comparing balls Similarity transformation PL homeomorphism between a triangle and a rectangle Basic open sets for disk and square Annulus A tube Ux × Y ⊂ Wx The topologist’s sine curve—two views Saturated open sets q −1 (U ) about [0] for [0, 1] and R Cylinder and torus as quotient spaces of the square Triangle as a quotient space of the square Expressing . on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Lawson, Terry, 1945– Topology : a geometric approach / Terry Lawson. 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